How do you find the number of roots for #f(x) = x^3 – 75x + 250# using the fundamental theorem of algebra?
1 Answer
The FTOA tells us that there are
Further investigation shows us that they are
Explanation:
The fundamental theorem of algebra tells you that any non-constant polynomial in one variable with Complex (possibly Real) coefficients has a zero which is a Complex (possibly Real) number.
A corollary of this, often stated as part of the FTOA is that a polynomial in one variable of degree
So in our example:
#f(x) = x^3-75x+250#
is of degree
Bonus
What else can we find out about these zeros?
By the rational root theorem, any rational zeros of
That means that the only possible rational zeros are positive and negative factors of
#+-1# ,#+-2# ,#+-5# ,#+-10# ,#+-25# ,#+-50# ,#+-125# ,#+-250#
We find:
#f(-10) = -1000+750+250 = 0#
So
#x^3-75x+250#
#=(x+10)(x^2-10x+25)#
#=(x+10)(x-5)(x-5)#
So the zeros are